Are curve physics possible?
Physics and simulations generally only affect mesh or particle objects, you often have to convert your curves to a mesh.
If you are using the builtin Blender tree generator Sapling Addon it even has an option to create an animated armature with wind animation, but even then, has far as I know you will have to convert it to mesh to work.

You may keep a copy of the original curve object in a separate scene or another layer in case you need to go back and change something.

It is however possible to use some basic soft body physics on Curve objects if it is well subdivided, although it does not work very reliably.

It seems most settings don't have any effect, except for the Self Collision options, as stated Quick Noodle Physics in Blender Tutorial Youtube Video

The trick is using a highly subdivided curve, there must be lots of actual vertex (display subdivisions are insufficient), then tweak the parameters under Soft Body Self Collision like *Ball Size and Stiffness and turning off Soft Body Goal.

Some other objects set to Collision Physics type must be present at the scene for interaction.